Powers of complete intersections: Graded Betti numbers and applications

Let I = (F_1,...,F_r) be a homogeneous ideal of the ring R = k[x(0),..., x(n)] generated by a regular sequence of type (d(1),...,d(r)). We give an elementary proof for an explicit description of the graded Betti numbers of I^s for any s >= 1. These numbers depend only upon the type and s. We then use this description to: (1) write H_{R/I^s}, the Hilbert function of R/I^s, in terms of H_R/I; (2) verify that the k-algebra R/I^s satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in P^n whose support is a complete intersection or a complete intersection minus a point.